Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 987545, 22 pages
Research Article

Moving Heat Source Reconstruction from the Cauchy Boundary Data

Programa de Engenharia Nuclear, COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 68509, CEP 21941-972, Rio de Janeiro, RJ, Brazil

Received 26 August 2010; Accepted 16 December 2010

Academic Editor: Francesco Pellicano

Copyright © 2010 Nilson C. Roberty and Marcelo L. S. Rainha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider the problem of reconstruction of an unknown characteristic transient thermal source inside a domain. By introducing the definition of an extended dirichlet-to-Neumann map in the time-space cylinder and the adoption of the anisotropic Sobolev-Hilbert spaces, we can treat the problem with methods similar to those used in the analysis of the stationary source reconstruction problem. Further, the finite difference 𝜃 scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic star-shape source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula, we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support.